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A nondiscrete random variable X is said to be absolutely continuous, or simply continuous, if its distribution function may be represented as
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where the function f (x) has the properties 1. f (x)
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It follows from the above that if X is a continuous random variable, then the probability that X takes on any one particular value is zero, whereas the interval probability that X lies between two different values, say, a and b, is given by
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CHAPTER 2 Random Variables and Probability Distributions
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EXAMPLE 2.4 If an individual is selected at random from a large group of adult males, the probability that his height X is precisely 68 inches (i.e., 68.000 . . . inches) would be zero. However, there is a probability greater than zero than X is between 67.000 . . . inches and 68.500 . . . inches, for example.
A function f (x) that satisfies the above requirements is called a probability function or probability distribution for a continuous random variable, but it is more often called a probability density function or simply density function. Any function f (x) satisfying Properties 1 and 2 above will automatically be a density function, and required probabilities can then be obtained from (8).
EXAMPLE 2.5
(a) Find the constant c such that the function
f (x)
is a density function, and (b) compute P(1 (a) Since f (x) satisfies Property 1 if c X 2).
0, it must satisfy Property 2 in order to be a density function. Now
` 3 ` 3
cx2 0
0 x 3 otherwise
and since this must equal 1, we have c (b) P(1 X
f (x) dx
2 30 cx dx
cx3 2 3 0
1 > 9. 2)
2 1 2 31 9 x dx
x3 2 27 1
8 27
1 27
7 27
In case f (x) is continuous, which we shall assume unless otherwise stated, the probability that X is equal to any particular value is zero. In such case we can replace either or both of the signs in (8) by . Thus, in Example 2.5, 7 P(1 X 2) P(1 X 2) P(1 X 2) P(1 X 2) 27
EXAMPLE 2.6 (a) Find the distribution function for the random variable of Example 2.5. (b) Use the result of (a) to find P(1 x 2).
(a) We have
F(x)
If x 0, then F(x) 0. If 0 x 3, then
f (u) du
F(x)
If x 3, then
30 f (u) du
x 1 2 30 9 u du
x3 27
F(x)
30 f (u) du
33 f (u) du
3 1 2 30 9 u du
33 0 du
Thus the required distribution function is
F(x)
0 x3 >27 1
x x x
0 3 3
Note that F(x) increases monotonically from 0 to 1 as is required for a distribution function. It should also be noted that F(x) in this case is continuous.
CHAPTER 2 Random Variables and Probability Distributions
(b) We have
2) 5 P(X 2) P(X 5 F(2) F(1) 13 7 23 5 27 27 27
as in Example 2.5.
The probability that X is between x and x P(x so that if X x
x is given by
f (u) du
x is small, we have approximately P(x X x x) f (x) x (10)
We also see from (7) on differentiating both sides that dF(x) dx f (x) (11)
at all points where f (x) is continuous; i.e., the derivative of the distribution function is the density function. It should be pointed out that random variables exist that are neither discrete nor continuous. It can be shown that the random variable X with the following distribution function is an example. 0 F(x) x 2 1 In order to obtain (11), we used the basic property d x f (u) du dx 3a f (x) (12) x 1 x 1 x 2 2
which is one version of the Fundamental Theorem of Calculus.
Graphical Interpretations
If f (x) is the density function for a random variable X, then we can represent y f(x) graphically by a curve as in Fig. 2-2. Since f (x) 0, the curve cannot fall below the x axis. The entire area bounded by the curve and the x axis must be 1 because of Property 2 on page 36. Geometrically the probability that X is between a and b, i.e., P(a X b), is then represented by the area shown shaded, in Fig. 2-2. The distribution function F(x) P(X x) is a monotonically increasing function which increases from 0 to 1 and is represented by a curve as in Fig. 2-3.
Fig. 2-2
Fig. 2-3
CHAPTER 2 Random Variables and Probability Distributions
Joint Distributions
The above ideas are easily generalized to two or more random variables. We consider the typical case of two random variables that are either both discrete or both continuous. In cases where one variable is discrete and the other continuous, appropriate modifications are easily made. Generalizations to more than two variables can also be made. 1. DISCRETE CASE. tion of X and Y by If X and Y are two discrete random variables, we define the joint probability funcP(X where 1. f (x, y)
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